Mathematics with Maple

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Mathematics with Maple A Comprehensive E-Book Harald Pleym Preface The main objective of these Maple worksheets, organized for use with all Maple versions from Maple 14, is to show how the computer

This interactive e-book can be used as a supplement to a standard textbook or on its own. All chapters in the worksheets summarize mathematical concepts and principles associated with each topic.

1 Mathematics with Maple A Comprehensive E-Book Harald Pleym Preface The main objective of these Maple worksheets, organized for use with all Maple versions from Maple 14, is to show how the computer algebra system Maple can be integrated as a valuable and powerful tool into the learning process of the material found in most mathematics textbooks at university levels. This interactive e-book can be used as a supplement to a standard textbook or on its own. All chapters in the worksheets summarize mathematical concepts and principles associated with each topic. There are more than 600 worked examples, presented in a dynamic and interactive form. These show how Maple can be used to solve problems and also reduce the amount of time spent in routine algebraic manipulation. With Maple there is a great opportunity for exploration, clarify concepts, trying different functions, zoom in on an area, visualize mathematical objects in two or

2 three dimensions, perform animations etc. I do assume that you have a basic familiarity with Maple, that you can start Maple, know how to enter commands, etc. If not, I recommend you to go through the Getting Started Guide This manual describes how to use the Standard Maple interface, which have two modes, document mode and worksheet mode. All worksheets in Mathematics with Maple are in worksheet mode, and all entering inputs are in text mode (maple input). Right-click the mouse and select Convert To 2-D Math if you want an input as you will find in a mathematics textbooks. If you need an overview of all Maple features and detailed help related to performing computations, creating plots, interactive documents etc., please consult the User Manual. You can also go to The Maple Portal. My own experience is that Maple's closeness to mathematical thinking, shortens the learning time of the syntax significantly. The body of text presents only the formal mathematics concepts needed for the worked solutions of the examples, which are familiar problems found in most mathematical textbooks, so that both the commands and the style of interaction with Maple can be mastered by students and others in a relatively short period of time. To get the most from these worksheets, you should work through the examples as they occur and in conjunction with the textbook you are using. Be aware that Maple can be used in several ways: Only as a solving engine. That means, you enter a mathematical expression which Maple subsequently execute, and you are freed from tedious computations, especially if you are not interested in how Maple gets the answer to a given command To present a method as a sequence of steps in much the same way as a textbook would, and illustrate the method with simple examples. The primary purpose here is to present intermediate steps to show you how to solve the problem. This is probably relevant in most of a student's homework solutions Unlike a math textbook you are free to execute larger and more realistic examples, and you can explore interactively how a solution of a problem depends on various parameters of the problem under study. The graphics capabilities are extremely helpful for visualizing the behavior of the system under investigation In the examples in the worksheets, I have tried to illustrate these methods of using Maple. The list of contents and all commands are hyperlinked to provide easy access to all chapters, and to the help pages for the commands used in the worksheets. Then you can assure that you are using the commands correctly, see Maple's own examples on the help pages, and get further help for the commands, Maple features, packages and more. You will also find hyperlinks to famous mathematicians on the web. A typical working session will begin by opening the contents section below. Select the link to the chapter and section you want to study. Open and press the Enter key repeatedly to execute in sequence the Maple commands that are already provided, or press the key to execute the entire worksheet. Hopefully these samples worksheets will serve as a guide when you prepare your own worksheets solutions to exercises and problems in math or other homework where Maple is used, and also inspire you to concentrate on exploring the many facets of mathematics. But, please bear in mind: You must do the thinking, Maple alone only does the calculations and move you more quickly into applications of problem solving. One of the beautiful qualities of Maple is that much can be done with few commands. Most of the examples in these worksheets involve no formal programming with Maple. I have included two

3 extra library packages, the calc package and the calcplot package. These are not part of the Maple software. The procedures in these packages are designed for visualizing concepts from calculus and advanced calculus, without any need for you to learn the subtleties of Maple's base commands in order to use the procedures. Many of the visualization procedures go far beyond Maple's built in visualization tools. My hope is that these worksheets will give you as a student easier access and additional help to make effective use of Maple, in classes, labs, homework and projects, and in future jobs. Maple allows the level of mathematical typesetting and also necessary word processing facilities needed for writing sets of exercises, homework and professional reports. Harald Pleym This Maple project is financially supported by Norwegian Non-Fiction Writers and Translators Association. Contents 1 Functions and Graphs 1.1 Functions and Inverse Functions 1.2 Graphs of and Functions 1.3 Polynomials and Algebraic Functions 1.4 Exponential and Logarithmic Functions 1.5 Trigonometric Functions and Their Inverses 1.6 Hyperbolic Functions and Their Inverses 2 Limits and Continuity 2.1 The Limit Concept 2.2 Theorems Involving Limits 2.3 Continuous Functions 3 Derivatives and Differentials 3.1 Average Rate of Change of a Function 3.2 Derivative of a Function 3.3 Differentiation Rules 3.4 The Chain Rule 3.5 Implicit Differentiation 3.6 Extreme Values and 4 Integration 4.1 Indefinite Integral 4.2 Area, Riemann Sums, and the Definite Integral 4.3 Integration Methods 4.4 Improper Integrals

4 Optimization 3.7 Rolle's Theorem and the Mean Value Theorem 3.8 Indeterminate Forms 5 Applications of Integrals 5.1 Area Between Two Curves 5.2 Volumes 5.3 Arch Length and Surface Area of Revolution 5.4 Mass, Moments and Centroids 5.5 Physical Applications 6 Vectors, Complex Numbers, and Linear Algebra 6.1 Vectors 6.2 Complex Numbers 6.3 Systems of Linear 6.4 Matrix Algebra 6.5 Determinants 6.6 Vector Spaces 6.7 Eigenvalues and Eigenvectors 6.8 Linear Transformations 6.9 Orthogonality and Least Squares 6.10 Symmetric Matrices and Quadratic Forms 6.11 Applications of Linear Algebra 7 Ordinary Differential 7.1 Differential and Initial Value Problems 7.2 First Order 7.3 Second Order Linear 7.4 Higher Order Linear 7.5 Applications 8 Conics, Polar Coordinates, and Parametric Curves 8.1 Conic Sections 8.2 Polar Coordinates 8.3 Parametric Curves 9 Curves and Surfaces in Space 9.1 Lines and Planes in Space 9.2 Vector-Valued Functions and Space Curves 10 Multivariable Functions and Partial Derivatives 10.1 Functions of Several Variables

5 9.3 Curvature and Acceleration 9.4 Cylinders and Quadratic Surfaces 9.5 Cylindrical and Spherical Coordinates 10.2 Limits and Continuity 10.3 Partial Derivatives 10.4 The Chain Rule 10.5 Directional Derivatives, The Gradient, Tangent Plane and Normal Line 10.6 Multivariable Optimization 11 Multiple Integration 11.1 Double Integrals 11.2 Iterated Double Integrals 11.3 Double Integrals in Polar Coordinates 11.4 Surface Area 11.5 Triple Integrals 11.6 Integration in Cylindrical and Spherical Coordinates 11.7 Change of Variables in Multiple Integrals 12 Vector Analysis 12.1 Vector Fields 12.2 Line Integrals 12.3 Independence of Path and Conservative Vector Fields 12.4 Green's Theorem 12.5 Surface Integrals 12.6 Stokes' Theorem 12.7 The Divergence Theorem 13 Systems of Differential 13.1 First Order Systems 13.2 Homogeneous Linear Systems 13.3 Nonhomogeneous Linear Systems 13.4 Linear Autonomous Systems 13.5 Nonlinear Autonomous Systems 14 Infinite Series 14.1 Sequences and Their Limits 14.2 Infinite Series and Convergence 14.3 Tests for Convergence 14.4 Power Series 14.5 Taylor and Maclaurin Series 14.6 Series Solutions of Differential 15 Fourier Series 15.1 Periodic Functions 15.2 Fourier Series 15.3 Fourier Sine and Cosine Series 15.4 Differentiation and Integration of Fourier Series 16 Transforms 16.1 Laplace Transforms 16.2 Fourier Integrals and Fourier Transforms 16.3 Z Transforms

6 17 Partial Differential 17.1 Partial Differential and Classification 17.2 Vibrating String and the Wave Equation 17.3 Heat Equation 17.4 Vibrating Membranes 17.5 Fourier Transform Method 17.6 Laplace Transform Method 18 Numerical Analysis 18.1 Newton's Method 18.2 Numerical Integration 18.3 Curve Fitting 18.4 Solution of Differential 18.5 Solution of Partial Differential 19 Linear Optimization 19.1 Optimization 19.2 Simplex Method 20 Logic, Combinatorics, and Statistics 20.1 Logic and Truth Tables 20.2 Combinatorics 20.3 Data Analysis 20.4 Probability and Probability Distributions 20.5 The Central Limit Theorem 20.6 Estimation 20.7 Tests of Statistical Hypotheses 20.8 Regression and Correlation 20.9 Analysis of Variance 21 Conformal Mappings 21.1 Conformal Mappings 22 Graphs and Trees 22.1 Graphs and Digraphs 22.2 Trees and Networks References Adams, R. A., Calculus, A Complete Course, Fifth Edition, Pearson Education, Anton, H., Bivens, I., Davis, S, Calculus, Early Transcendentals, 7th Edition, John Wiley & Sons, Inc., Anton, H., Busby, R. C., Contemporary Linear Algebra, 7th Edition, John Wiley & Sons, Inc., Anton, H., Rorres, C., Elementary Linear Algebra, Applications Version, Seventh Edition, John Wiley & Sons, Inc., 1994

7 Asmar, N. H., Partial Differential with Fourier Series and Boundary Value Problems, Second Edition, Pearson Education, Inc., Boyce, W. E., DiPrima, R. C., Elementary Differential and Boundary Value Problems, Fifth Edition, John Wiley & Sons, Inc., Bugl, P., Differential, Matrices and Models, Prentice Hall, Inc., Carnahan, B., Luther, H. A, Wilkes, J. O, Applied Numerical Methods, John Wiley & Sons, Inc., Edwards, C. H., Penney, D. E., Calculus 6/e, Early Transcendentals-Matrix Version, Prentice-Hall, Inc., Edwards, C. H., Penney, D. E., Differential and Boundary Value Problems, Computing and Modeling, 3E, Prentice Hall, Inc., Edwards, C. H., Penney, D. E., Differential & Linear Algebra, Prentice Hall, Inc., Finney, R. L., Weir, M. D., Giordano, F. G., Thomas' Calculus Early Transcendentals, Tenth Edition, Addison Wesley, Greenberg, M. D., Differential & Linear Algebra, Prentice Hall, Inc., Hogg, R. V., Probability and Statistical Inference, Fourth Edition, MacMillan Publishing Company, James, G., Advanced Modern Engineering Mathematics, Third Edition, Pearson Education Limited, Kreyszig, E., Advanced Engineering Mathematics, 8th ed., John Wiley & Sons, Inc., Lay, D. C., Linear Algebra and its Applications, Third Edition, Pearson Education Inc., Logan, D. J, Applied Partial Differential, Springer Verlag, Inc., Maplesoft, Maple 12, The Essential Tool for Mathematics and Modeling, User Manual, Waterloo Maple Inc., Maplesoft, Maple 12, The Essential Tool for Mathematics and Modeling, Introductory Programming Guide, Waterloo Maple Inc., Maplesoft, Maple 12, The Essential Tool for Mathematics and Modeling, Advanced Programming Guide, Waterloo Maple Inc., Matthews, P. C., Vector Calculus, Springer Verlag, Inc., Mathews, J. H., Numerical Methods for Mathematics, Science, and Engineering, Second Edition, Prentice Hall International Editions Verlag, Inc., Rosen, K. H., Discrete Mathematics and its Applications, Third Edition, McGraw-Hill, Inc., Walpole, R. E., Myers, R. M., Probability and Statistics for Engineers and Scientists, Sixth Edition, Prentice Hall International, Inc., Wylie, C. R., Barrett, C. L., Advanced Engineering Mathematics, Sixth Edition, McGraw- Hill, Inc., 1995.

Maple in Calculus. by Harald Pleym. Maple Worksheets Supplementing. Edwards and Penney. CALCULUS 6th Edition Early Transcendentals - Matrix Version

Maple in Calculus. by Harald Pleym. Maple Worksheets Supplementing. Edwards and Penney. CALCULUS 6th Edition Early Transcendentals - Matrix Version Maple in Calculus by Harald Pleym Maple Worksheets Supplementing Preface Edwards and Penney CALCULUS 6th Edition Early Transcendentals - Matrix Version These worksheets provide a comprehensive Maple supplement